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Helen's Conjecture

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

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Marbles

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

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More Marbles

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

Where Can We Visit?

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This problem offers all students opportunities to explore fundamental ideas about number theory in a simple context. They are encouraged to explore, conjecture, generalise and justify. There are opportunities for older students who are familiar with algebraic manipulation or modulo arithmetic to produce rigorous proofs.

Possible approach

You need a set of number cards 1-100, with the multiples of 5 removed. Shuffle the cards in front of the class and hand out one to each student. Do not tell them that the multiples of 5 are missing!
Ask each student to turn to their neighbour and work out how to get from one number to the other and back again, using only these two operations:
$\times 2$ and $-5$

For example if the two numbers are 21 and 54 the chains could be:

21, double, 42, take five, 37, take five, 32, take five, 27, double, 54
and
54, take lots of fives, 9, double, 18, take five, 13, double, 26, take five, 21.

Pairs that are proving difficult to connect could be written on the board and offered as a challenge for the whole class to solve. Everyone should be able to arrive at their partner's number!

Finally challenge the class to get to your number (which should be a carefully chosen multiple of five). You may wish to offer a prize......
Once the class give up, ask them to explain why it is impossible.

Display the 1-100 grid, choose 42 as your starting number and explain that by using the operations above we are going to try to visit all the numbers on the grid.
Demonstrate how the numbers are crossed out as they are visited.
Ask the students to predict what will happen.
Will they be able to visit every number on the grid at least once?

Hand out this 1-100 grid and allow some time for students to work in pairs to check their predictions.
Bring the students together to link their ideas to the findings from the earlier exercise.
What would have happened if they had started on a different number?
Can they explain their results?

Ask if they think they will get the same sort of results with other pairs of operations.
You may wish to suggest families of pairs of operations for them to explore. Eg:
x3 and -5
x4 and -5
x5 and -5...
or
x5 and -2
x5 and -3
x5 and -4...
or they can try some families of their own choosing.

Hand out plenty of the 1-100 grids and ask students to work in pairs or small groups and make a display of their results.

Can they explain their findings and use these to begin to make predictions about other pairs of operations? Encourage them to justify their predictions.

You might find it useful to see if they can identify the pair of operations that produced the patterns in the three grids below. They can choose from either:
$\times 3$ and $-6$,
or
$\times 6$ and $-3$
See if they can spot which is which, and if the starting number makes a difference.


Key questions

What happens to multiples of 5 when they are doubled?
What about numbers that are 1 more, 2 more, 3 more and 4 more than a multiple of 5?

What happens to multiples of 5 when 5 is subtracted from them?
What about numbers that are 1 more, 2 more, 3 more and 4 more than a multiple of 5?

Possible extension

Students may be interested in this introductory reading on Modular Arithmetic.

Students could have a go at Take Three from Five which has a similar underlying structure. Can they use their insights from the previous problem to solve this challenge?

Possible support

Most students will have little difficulty with doubling or subtracting 5, but may find it more difficult to spot the patterns involved. Using coloured arrows to represent repeated subtractions of 5 on various 1-100 grids (each starting from a different number) may help them to see the patterns emerge.